We study a robust extensible bin packing problem, under a budget uncertainty model where item sizes lie in the intersection of a box with a one-norm ball. We design an algorithm based on the column-and-constraint generation framework, which alternates between solving a master robust bin-packing problem with a finite uncertainty set and solving a separation problem. We show that the separation is strongly NP-hard given solutions to the continuous relaxation of the master problem. Then, focusing on the separation problem for the integer master problem, we show that this problem becomes a special case of the continuous convex knapsack problem, which is known to be weakly NP-hard. Next, we prove that our special case remains NP-hard. We develop a pseudo-polynomial dynamic program (DP) and a fully polynomial-time approximation scheme (FPTAS) for our special case whose running times match those of the binary knapsack problem. Finally, our computational study shows that the DP can be significantly more efficient in practice compared with solving the problem with specially ordered set (SOS) constraints using advanced mixed-integer programming (MIP) solvers. Our experiments also demonstrate the application of our separation problem method to solving the robust extensible bin packing problem, including separating based on approximate master solutions in intermediate iterations. We further enhance the algorithm by proposing custom symmetry breaking inequalities, valid when bins have identical costs per unit of excess capacity. Finally, a case-study, based on real elective surgery data, demonstrates the potential advantage of our model compared with the actual schedule and optimal nominal schedules.

翻译：暂无翻译